Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



On the location of critical points of polynomials

Authors: Branko Curgus and Vania Mascioni
Journal: Proc. Amer. Math. Soc. 131 (2003), 253-264
MSC (2000): Primary 30C15; Secondary 26C10
Published electronically: June 3, 2002
MathSciNet review: 1929045
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Abstract: Given a polynomial $p$ of degree $n \geq 2$ and with at least two distinct roots let $Z(p) = \{z : p(z) = 0\}$. For a fixed root $\alpha \in Z(p)$ we define the quantities $\omega(p,\alpha) := \min\bigl\{\vert\alpha - v\vert : v \in Z(p)\setminus \{\alpha\} \bigr\}$and $\tau(p,\alpha) := \min\bigl\{\vert\alpha - v\vert : v \in Z(p')\setminus \{\alpha\} \bigr\}$. We also define $\omega(p)$ and $\tau(p)$ to be the corresponding minima of $\omega(p,\alpha)$ and $\tau(p,\alpha)$ as $\alpha$ runs over $Z(p)$. Our main results show that the ratios $\tau(p,\alpha)/\omega(p,\alpha)$ and $\tau(p)/\omega(p)$ are bounded above and below by constants that only depend on the degree of $p$. In particular, we prove that $(1/n)\omega(p)\leq\tau(p)\leq\bigl(1/2\sin(\pi/n)\bigr)\omega(p)$, for any polynomial of degree $n$.

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Additional Information

Branko Curgus
Affiliation: Department of Mathematics, Western Washington University, Bellingham, Washington 98225

Vania Mascioni
Affiliation: Department of Mathematics, Western Washington University, Bellingham, Washington 98225

Keywords: Roots of polynomials, critical points of polynomials, separation of roots
Received by editor(s): July 10, 2001
Received by editor(s) in revised form: September 4, 2001
Published electronically: June 3, 2002
Communicated by: N. Tomczak-Jaegermann
Article copyright: © Copyright 2002 American Mathematical Society